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 Issue A&A Volume 569, September 2014 A47 15 Planets and planetary systems https://doi.org/10.1051/0004-6361/201423966 18 September 2014

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Appendix A: Appendix A

Equation (6) contains two factors raised to the power of − 1, which may take the value zero. The first is sini, which implies that an integration of the formula cannot be performed, if the inclination is zero. The other factor is more benign. It can be written as where a and p are the semi-major axis and semilatus rectum of the projectile, and the heliocentric distance ρ of the target is expressed by Eq. (7). We use for the semilatus rectum of the target. The reciprocal of the factor under consideration is The non-singular nominator is easily dealt with by numerical integration, so we need to concentrate only on the integral, (A.1)where ν is the true anomaly of the target. The denominator can be written as , where q and Q are the perihelion and aphelion distances of the projectile, respectively. The integrand may thus become singular, only if there is a value of ν such that ρ = q or ρ = Q. In the present case, we are only interested in the occurrence of ρ = q, when the perihelion distance of the projectile is spanned by the target’s orbit. By νc we denote the positive true anomaly of the target, for which ρ = q. If | ν | <νc, collisions are excluded, and we may neglect this part of the integral, considering only (A.2)The non-singular factor can again be disregarded, so we concentrate on the expression We note that the above manipulation fails in case et = 0. Hence, the integration cannot be performed, if the target orbit is circular with at = q. Excluding this case. the non-singular nominator can again be disregarded, so we concentrate on the integral, Substituting (A.3)the integral is transformed into which is proper, when cos2(νc/2) ≠ 1, or νc ≠ 0. Our method to integrate the Wetherill formula starts by flagging an error, if i = 0 or qt = q (i.e., νc = 0). Otherwise, we either have qt>q, in which case a Simpson rule integration of the Wetherill expression is performed without problems (as long as Qt<Q), or qt<q, in which case the Simpson rule integration is easily carried out using the variable substitution of Eq. (A.3). We finally note that a different analytic proof of the regularity of the integral was presented by Wetherill (1967).

In summary, we have found the following cases, when the Wetherill integral diverges due to singularities:

• co-planar orbits of the target and projectile (i = 0);

• circular target orbit at the perihelion distance of the projectile;

• elliptic target orbit with the same perihelion distance as the projectile.

Although we did not consider it explicitly here, the third case has an analog for a different type of projectile orbit, if the two aphelion distances are equal.

Appendix B: Appendix B

Valsecchi et al. (2003) present an analytic theory of close encounters by extending the work of Öpik (1976), in which the effects of a close encounter of a small body with a planet moving in a circular heliocentric orbit is described making use of two b-plane coordinates, namely, ξ and ζ. From these quantities, the latter is related to the time difference with which the small body and the planet arrive at their closest approach, while the modulus of the former is the MOID (Valsecchi et al. 2003, Sect. 3.1). It is clear that an encounter within a given distance d can only take place if the MOID is smaller than d; that is, if | ξ | ≤ d.

Building on that theory and on refined expressions given in Valsecchi (2006), it is possible to express small displacements of ξ and ζ from the origin in terms of the partial derivatives of ξ and ζ with respect to the orbital elements a,e,i,ω,Ω, and f of the small body at collision with the planet (neglecting the gravitational influence of the latter). Valsecchi et al. (2005) give a suitable procedure for this computation.

In terms of these derivatives, keeping a,e,i constant and d suitably small, we have: Appendix C: Appendix C

The target has a circular orbit around the Sun with i = 0, which is situated at a distance of 1 AU. We denote here its collisional radius by Δ. Consider a projectile orbit with ω = 0. Thus, its perihelion point coincides with the ascending node. We also assume that so the perihelion point is situated on the surface of the target ring, where this cuts the reference plane on the side facing the Sun.

If the projectile inclination is i, we can calculate the distance of the projectile from the target orbit at any true anomaly f, since we have (C.1)for the heliocentric distance expressed in AU, using p = a(1 − e2) for the semilatus rectum, also expressed in AU. Using cylindrical coordinates (s,ϑ,z) (see Fig. C.1), we get (C.2)and we can write the square of the distance between the projectile and the target orbit as (C.3)Concerning the function t(f), we know that t(0) = Δ2 and the first derivative, t′(0) = 0, since the motion of the projectile is orthogonal to the direction to the target ring at perihelion. Therefore, whether or not the projectile orbit enters into the ring to first order is determined by the value of the second derivative, t′′(0). We now set out to calculate this. Fig. C.1 Geometric illustration for orbits with perihelia close to the target ring. The z axis is perpendicular to the orbital plane of the target, and a cross-section of the ring is shown at unit distance from the Sun, including its radius Δ. Open with DEXTER

From the above equations, we get which upon differentiation yields (C.4)We easily verify that t′(0) = 0. This equation can be written as where and B is the expression in curly brackets, so we have Obviously, A(0) = 0, so we get We easily find that which, like p, is positive definite, so the sign of t′′(0) is the same as the sign of There is, hence, a unique inclination ic, for which B(0) = 0, given by (C.5)Since we have p = q′(1 + e) = (1 − Δ)(1 + e), we obtain and since Δ ≪ 1, the final result is (C.6)We easily verify that B(0) < 0 at i<ic, while B(0) > 0 at i>ic. Thus, for the particular orbit that we have chosen, t has a maximum for f = 0 in the first case and a minimum in the second case. In conclusion, for the lowest inclinations (i<ic), the orbit enters into the target ring on both sides of perihelion, while it just touches the ring at perihelion for higher inclinations. If we decrease q slightly in the first case, we get a double ring crossing, and in the second case, we get no crossing at all. If, on the other hand, we increase q slightly, we always get a single ring crossing.